
theorem LX1:
for L being add-associative right_zeroed right_complementable
            right-distributive right_unital non empty doubleLoopStr,
    a being Element of L holds a|L = a * 1_.(L)
proof
let L be add-associative right_zeroed right_complementable
            right-distributive right_unital non empty doubleLoopStr,
    a be Element of L;
now let x be object;
  assume x in NAT;
  then reconsider i = x as Element of NAT;
  per cases;
  suppose A: i = 0;
    thus (a * 1_.(L)).x = a * (1_.(L)).i by POLYNOM5:def 4
             .= a * 1.L by A,POLYNOM3:30
             .= a
             .= (a|L).x by Th28,A;
    end;
  suppose A: i > 0;
    thus (a * 1_.(L)).x = a * (1_.(L)).i by POLYNOM5:def 4
             .= a * 0.L by A,POLYNOM3:30
             .= (a|L).x by Th28,A;
    end;
  end;
hence thesis by FUNCT_2:12;
end;
